Piecewise rational approximation to continuous functions with characteristic singularities

Let Hvp[a,b] be the class of continuous functions in the interval [a,b], which admit analytic continuation to Hp in the subintervals of a v-subdivition of [a,b]. Let Sn,v[a,b] be the set of piecewise rational functions which have pieces of degree n and no more than v breakpoints. The paper deals with the construction of fn,v ∈ Sn,y[a,b] close to a given f ∈ Hvp[a,b], to estimate the order of the least Lp(w) distance from f to Sn,v[a,b]. It is concluded that the piecewise rational approximation is better for Hvp[a,b] than the rational one. This theory extends results of Gonchar (1967) on the rational approximation to f ∈ H0∞[−1,1], and can be applied to study the asymptotic behavior of piecewise rational methods to solve a singular identification problem associated with the equation w(2) + (f + λ)w = 0, in terms of the error equation criterion.